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The original Broad Learning System (BLS) on new added nodes and its existingefficient implementation both assume the ridge parameter lambda -> 0 in theridge inverse to approximate the generalized inverse, and compute thegeneralized inverse solution for the output weights. In this paper, we proposetwo ridge solutions for the output weights in the BLS on added nodes, wherelambda -> 0 is no longer assumed, and lambda can be any positive real number.One of the proposed ridge solutions computes the output weights from theinverse Cholesky factor, which is updated efficiently by extending the existinginverse Cholesky factorization. The other proposed ridge solution computes theoutput weights from the ridge inverse, and updates the ridge inverse byextending the Greville's method that is a classical tool to compute thegeneralized inverse of partitioned matrices. For the proposed efficient ridgesolution based on the inverse Cholesky factor, we also develop anotherimplementation that is numerically more stable when the ridge parameter lambdais very small. The proposed ridge solution based on the ridge inverse and thenumerically more stable implementation of the proposed efficient ridge solutionrequire the same complexity as the original BLS and the existing efficient BLS,respectively. Moreover, the speedups of the proposed efficient ridge solutionto the original BLS and the existing efficient BLS are 3 and more than 1.67respectively, when the computational complexities for each update are compared,and the speedups are 1.99 - 2.52 and 1.31 - 1.58, respectively, when the totaltraining time is compared by numerical experiments. On the other hand, ournumerical experiments show that both the proposed ridge solutions for BLSachieve better testing accuracies than the original BLS and the existingefficient BLS.

Two Ridge Solutions for the Incremental Broad Learning System on Added Nodes